Problem: Simplify; express your answer in exponential form. Assume $t\neq 0, z\neq 0$. $\dfrac{{(t^{5}z^{-1})^{-3}}}{{(tz^{-3})^{-2}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(t^{5}z^{-1})^{-3} = (t^{5})^{-3}(z^{-1})^{-3}}$ On the left, we have ${t^{5}}$ to the exponent ${-3}$ . Now ${5 \times -3 = -15}$ , so ${(t^{5})^{-3} = t^{-15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(t^{5}z^{-1})^{-3}}}{{(tz^{-3})^{-2}}} = \dfrac{{t^{-15}z^{3}}}{{t^{-2}z^{6}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{-15}z^{3}}}{{t^{-2}z^{6}}} = \dfrac{{t^{-15}}}{{t^{-2}}} \cdot \dfrac{{z^{3}}}{{z^{6}}} = t^{{-15} - {(-2)}} \cdot z^{{3} - {6}} = t^{-13}z^{-3}$